# topology

1. ### Seeking sources: Mathematics of knots

Hello everyone, First of all, I posted this thread here because I really wasn't too sure where else to. I suppose it could have gone in the "Analysis, Topology and Differential Geometry" section, but this isn't exactly university level... well, I'll explain. I apologise if the thread would...
2. ### Question about ordered topology

This is a Munkres's problem: Let Y be an ordered set with the ordered topology and f,g functions from X to Y, show that the set\{x:f(x)\leq g(x)\} is closed. I can show this, for intance, when Y=R(real numbers set) but I can't prove this if Y is just an ordered set because it may have...
3. ### Metric Topology

Hello everybody I want to show that the quotient operation \mathbb{R}^2 \longrightarrow \mathbb{R}is continuous in \mathbb{R} with the metric d(a,b) = |a-b| on \mathbb{R} and the metric on \mathbb{R}^2 given by the equation \rho((x,y),(x_0,y_0)) = \max\{|x-x_0|,|y-y_0|\} Start by given...
4. ### Basic point-set topology question (closure of rings of sets under union)

Edit: I think I figured it out (immediately after going to bed, of course). Take the symmetric difference of A and B, and also take their intersection. Then take the symmetric difference of their symmetric difference and their intersection. I haven't verified this, but just thinking about it...
5. ### Topology

Problem: Let X\subseteq \mathbb{R} be compact. Prove or disprove that \mathcal{C}\left[X,\mathbb{R}\right] is separable in general.
6. ### Cauchy Sequence in a Metric Space

Problem: Let {x_{i}} be a Cauchy sequence in a metric space (M,D). Let A={x_{1}, x_{2}, x_{3}, ...}. Suppose that {x_{i}} doesn't converge in M. Prove that A is a closed subset of (M,D). What I have done: So we know {x_{i}} is Cauchy, so let \epsilon>0 be given. Then there exists a...
7. ### Basic topology

Hello, I have the following problem: Prove that the set of limit points E' of a subset E is a closed set. Help please, Thank you,
8. ### Topology questions

Can someone please help with these questions by using examples: 1. Does there exist a continuous function from (0,1) onto [0,1] ? 2 Does there exist a continuous function from (-1,1) onto R ? 3 Does there exist a continuous function from R onto (-1,1) ? 4 Does...
9. ### Topology questions

I need help with these questions 1. Does there exist a continuous function from (0,1) onto [0,1] 2. Does there exist a continuous function from (-1,1) onto R 3. Does there exist a continuous function from R onto (-1, 1) 4. Does there exist a continuous function from R onto Q 5. Does there...
10. ### Topology question

Hi, I would appreciate some guiding with this question. Thanks. Let X be some topological space and Y a compact, Hausdorff topological space. Let f:X \to Y and define \Gamma_f = \{(x, f(x)) : x \in X\}. Show that if \Gamma_f is a closed subset of X \times Y with the product topology...
11. ### Quesion on topology relating to produc spaces

Question on topology relating to product spaces Hello: I have what may be a silly question because did not find it in the books I looked for it and is this: Is the product of closed sets closed in the product topology? The answer seems to me to be "yes".
12. ### Fort Space Topology

For the Fort space topology, let (a_{n}) be a sequence in X, such that the set of the sequence, \{a_{n}:n\in\mathbb{N}\}, is infinite. Using the definition of convergence in topological space, prove that (a_{n}) has a subsequence which converges to p. I'm struggling to make the...
13. ### Properties of the product topology

Hello friends. I am doing an independent study course, and it is a bit of the Moore method style. So, right now I am studying product topology and have come up with some conjectures. I have "proof" for all of them but would appreciate (no need for proof if you don't want) if someone could...
14. ### Product Topology

Hello everyone. I have the following problem. The quote says to consider that if X were second countable that every open base must have a countable subset which is also an open base. But, I don't understand why this doesn't work. Clearly we have that G_j=\prod_{k\in[0,1]}E_k where...
15. ### [SOLVED] Topology

Let (A_\lambda)_{\lambda\in\mathbb{N}} be a collection of open sets of real numbers. Suppose F\subset\mathbb{R} is such that \overline{F\cap A_\lambda}=F \forall \lambda\in\mathbb{N}. Show that \overline{F\cap\bigcap_{\lambda\in\mathbb{N}}A_\lambda}=F I've managed to prove that...
16. ### Fort Space Topology

Does anybody know which sets are both open and closed, and which sets are neither open nor closed for the Fort Space Topology?
17. ### Munkres topology Second Edition

Please solve question no. 9 and 13 for section 18 on page 112. Question no. 9 and 10 for section 13 on page 152. solve question no.11 on page 129 for section 20
18. ### Product Topology

Please solve question no. 2 in the attached assignment.
19. ### Basic topology question

This is relatively easy, but it requires an astute observation. You use this quite a bit later in topology so I guess it's a good challenge question. NOTE: If you have already seen the solution to this, please don't ruin the fun! This is aimed at people who haven't seen a lot of topology...
20. ### [Topology] Connected Spaces

I realy need some help in the following questions: 1. Let X be a topological space and let A be subgroup of X that contains one and only point from each connected component of X. Prove that X/A is connected. 2. Let X=R-{0} be a subspace of R. Prove that for each n=1,2,3,4... - X^n is...